Gaussian measure

In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space Rⁿ, closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces. Gaussian measures are named after the German mathematician Carl Friedrich Gauss. One reason why Gaussian measures are so ubiquitous in probability theory is the Central Limit Theorem. Loosely speaking, it states that if a random variable X is obtained by summing a large number N of independent random variables of order 1, then X is of order $\sqrt{N}$ and its law is approximately Gaussian.

Definitions

Let n ∈ N and let B₀(Rⁿ) denote the completion of the Borel σ-algebra on Rⁿ. Let λⁿ : B₀(Rⁿ) → [0, +∞] denote the usual n-dimensional Lebesgue measure. Then the standard Gaussian measure γⁿ : B₀(Rⁿ) → [0, 1] is defined by

$\gamma^{n} (A) = \frac{1}{\sqrt{2 \pi}^{n}} \int_{A} \exp \left( - \frac{1}{2} \| x \|_{\mathbb{R}^{n}}^{2} \right) \, \mathrm{d} \lambda^{n} (x)$

for any measurable set A ∈ B₀(Rⁿ). In terms of the Radon–Nikodym derivative,

$\frac{\mathrm{d} \gamma^{n}}{\mathrm{d} \lambda^{n}} (x) = \frac{1}{\sqrt{2 \pi}^{n}} \exp \left( - \frac{1}{2} \| x \|_{\mathbb{R}^{n}}^{2} \right).$

More generally, the Gaussian measure with mean μ ∈ Rⁿ and variance σ² > 0 is given by

$\gamma_{\mu, \sigma^{2}}^{n} (A) := \frac{1}{\sqrt{2 \pi \sigma^{2}}^{n}} \int_{A} \exp \left( - \frac{1}{2 \sigma^{2}} \| x - \mu \|_{\mathbb{R}^{n}}^{2} \right) \, \mathrm{d} \lambda^{n} (x).$

Gaussian measures with mean μ = 0 are known as centred Gaussian measures.

The Dirac measure δ_μ is the weak limit of $\gamma_{\mu, \sigma^{2}}^{n}$ as σ → 0, and is considered to be a degenerate Gaussian measure; in contrast, Gaussian measures with finite, non-zero variance are called non-degenerate Gaussian measures.

Properties of Gaussian measure

The standard Gaussian measure γⁿ on Rⁿ

• is a Borel measure (in fact, as remarked above, it is defined on the completion of the Borel sigma algebra, which is a finer structure);
• is equivalent to Lebesgue measure: $\lambda^{n} \ll \gamma^{n} \ll \lambda^{n}$, where $\ll$ stands for absolute continuity of measures;
• is supported on all of Euclidean space: supp(γⁿ) = Rⁿ;
• is a probability measure (γⁿ(Rⁿ) = 1), and so it is locally finite;
• is strictly positive: every non-empty open set has positive measure;
• is inner regular: for all Borel sets A,

$\gamma^{n} (A) = \sup \{ \gamma^{n} (K) | K \subseteq A, K \mbox{ is compact} \},$

so Gaussian measure is a Radon measure;

• is not translation-invariant, but does satisfy the relation

$\frac{\mathrm{d} (T_{h})_{*} (\gamma^{n})}{\mathrm{d} \gamma^{n}} (x) = \exp \left( \langle h, x \rangle_{\mathbb{R}^{n}} - \frac{1}{2} \| h \|_{\mathbb{R}^{n}}^{2} \right),$

where the derivative on the left-hand side is the Radon–Nikodym derivative, and (T_h)_∗(γⁿ) is the push forward of standard Gaussian measure by the translation map T_h : Rⁿ → Rⁿ, T_h(x) = x + h;

• is the probability measure associated to a normal probability distribution:

$Z \sim \mathrm{Normal} (\mu, \sigma^{2}) \implies \mathbb{P} (Z \in A) = \gamma_{\mu, \sigma^{2}}^{n} (A).$

Gaussian measures on infinite-dimensional spaces

It can be shown that there is no analogue of Lebesgue measure on an infinite-dimensional vector space. Even so, it is possible to define Gaussian measures on infinite-dimensional spaces, the main example being the abstract Wiener space construction. A Borel measure γ on a separable Banach space E is said to be a non-degenerate (centered) Gaussian measure if, for every linear functional L ∈ E^∗ except L = 0, the push-forward measure L_∗(γ) is a non-degenerate (centered) Gaussian measure on R in the sense defined above.

For example, classical Wiener measure on the space of continuous paths is a Gaussian measure.

See also

• Besov measure, a generalisation of Gaussian measure
• Cameron–Martin theorem